1. Field of the Invention
This invention relates to a method for energizing a shunt capacitor, more particularly to a voltage peak switch closing method for shunt capacitor energization.
2. Description of the Related Art
In an alternating voltage system, the magnitude of an instantaneous voltage varies with time. Since the phase of capacitor current leads the capacitor voltage by 90 degrees in steady state, the current peak appears at voltage zero while the current zero appears at voltage peak. Therefore, shunt capacitor energization often encounters the problem of inrush current and transient overvoltage. The voltage surge even occurs at remote locations with a capacitive load.
FIG. 1 shows a capacitor switching circuit, while FIG. 2 shows capacitor voltage and current waveforms of the circuit of FIG. 1 in steady state. In order to analyze the capacitor switching circuit of FIG. 1, the Kirchhoff's Voltage Law (KVL) and Laplace Transformation were used to deal with the problem of time domain switching circuit with initial value. With a sinusoidal ac supply voltage EQU V(t)=V.sub.m sin (.omega.t+.alpha.)
where V.sub.m is the peak value of the ac voltage. The source voltage indicates a phasor varying at the supply frequency .omega.. The inclusion of the arbitrary phase angle .alpha. permits closing of the switch (SW) at any instant in the voltage cycle. When the switch (SW) is closed, the equation expressed in terms of the current is EQU L[di(t)/dt]+(1/c).intg.i(t)dt=V(t)
The current equation and voltage equation in terms of the Laplace transform are EQU L[sI(s)-I(0)]+I(s)/sC+Vc(0)/s=V(s) EQU V(s)=V.sub.m {[.omega. cos .alpha./(s.sup.2 +.omega..sup.2)]+[s sin .alpha./(s.sup.2 +.omega..sup.2)]}
where I(0) and Vc(0) are the initial values of the inductor current and the capacitor voltage respectively. In this circuit, the operational solution for current is ##EQU1## where .omega..sub.n is the natural frequency of the switching circuit. ##EQU2## The first term on the right-hand side of equation (1) can be rewritten as follows: ##EQU3## where n is the per unit natural frequency. EQU n=.omega..sub.n /.omega.
Taking the inverse Laplace transform ##EQU4## where I.sub.m and V.sub.m are the peak values of the capacitor current and voltage in steady state respectively. EQU I.sub.m =[(n.sup.2 /(n.sup.2 -1))(V.sub.m /X.sub.c)=V.sub.m /X.sub.c
Equation (1) can be evaluated with the aid of equation (2). EQU i(t)=I.sub.m cos (.omega.t+.alpha.)+[I(0)-I.sub.m cos .alpha.] cos .omega..sub.n t+n[I.sub.m sin .alpha.-(V.sub.c (0)/X.sub.c)]sin .omega..sub.n t (3)
Similarly, the capacitor voltage is derived as follows: EQU (d.sup.2 v.sub.c (t)/dt.sup.2)+(1/LC)v.sub.c (t)=(1/LC)v(t)
By straight forward transform manipulation and inverse transformation, we get the instantaneous voltage: EQU v.sub.c (t)=V.sub.m sin (.omega.t+.alpha.)+[V.sub.c (0)-V.sub.m sin .alpha.]cos .omega.t+(1/n)[X.sub.c I(0)-V.sub.m cos .alpha.]sin .omega..sub.n t (4)
Equations (3) and (4) represent the time domain response of the capacitor current and voltage in the switching circuit shown in FIG. 1. The first term is the fundamental frequency component. The second and third terms represent the oscillatory components with circuit natural frequency .omega..sub.n.
The magnitude of the oscillatory components is a function of the system voltage, capacitor trapped voltage, inductor current and the switch closing time.
In general, capacitors are initially discharged [V.sub.c (0)=0]. The worst case occurs at the time when a discharged capacitor is energized at the instant of voltage peak which results in a transient overvoltage near twice the normal peak voltage. If a capacitor is trapped with peak voltage [V.sub.c (0)=+V.sub.m ], the worst case occurs when the capacitor is energized at the instant of voltage peak with opposite polarity. It brings a transient overvoltage near three times the normal peak voltage.
For ideal switching, the oscillatory components of current in equation (3) [voltage in equation (4)] must be zero. This can happen only when the following two conditions are simultaneously satisfied. EQU I(0)=I.sub.m cos .alpha. (A) EQU V.sub.c (0)=V.sub.m sin .alpha. (B)
A voltage zero switch closing method has been proposed for shunt capacitor energization. For a discharged capacitor, from the condition (B), the closing time is selected at voltage zero (i.e., .alpha.=0.degree. or .alpha.=180.degree.), so that the condition (A) will be EQU I(0)=.+-.I.sub.m (5)
Equation (5) means that the initial value of inductor current must be equal to the peak value of the fundamental frequency capacitor current. However, it is difficult and expensive to accomplish this technique up to now. Therefore, the discharged capacitor energized by the voltage zero closing method will produce a high frequency oscillatory component due to equation (5) not being satisfied, as shown in FIGS. 3A and 3B.
The equations of capacitor current and voltage are shown in FIG. 4. The magnitude of the high frequency oscillatory current component is the same as the fundamental frequency component. The magnitude of the high frequency oscillatory voltage component is related to the fraction of the natural frequency of the switching circuit. Lower natural frequency results in a larger high frequency oscillatory component.
It should be noted that the transient overvoltage as described above damages not only the capacitor, but also the other components that are connected to the capacitor.